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Properties

Label 12-51e12-1.1-c1e6-0-1
Degree $12$
Conductor $3.096\times 10^{20}$
Sign $1$
Analytic cond. $8.02610\times 10^{7}$
Root an. cond. $4.55731$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s + 3·4-s + 3·5-s + 3·7-s − 2·8-s − 9·10-s + 9·11-s + 9·13-s − 9·14-s + 3·16-s + 9·19-s + 9·20-s − 27·22-s + 9·23-s − 3·25-s − 27·26-s + 9·28-s + 6·29-s + 24·31-s − 6·32-s + 9·35-s − 3·37-s − 27·38-s − 6·40-s + 18·41-s + 27·44-s − 27·46-s + ⋯
L(s)  = 1  − 2.12·2-s + 3/2·4-s + 1.34·5-s + 1.13·7-s − 0.707·8-s − 2.84·10-s + 2.71·11-s + 2.49·13-s − 2.40·14-s + 3/4·16-s + 2.06·19-s + 2.01·20-s − 5.75·22-s + 1.87·23-s − 3/5·25-s − 5.29·26-s + 1.70·28-s + 1.11·29-s + 4.31·31-s − 1.06·32-s + 1.52·35-s − 0.493·37-s − 4.37·38-s − 0.948·40-s + 2.81·41-s + 4.07·44-s − 3.98·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 17^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 17^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{12} \cdot 17^{12}\)
Sign: $1$
Analytic conductor: \(8.02610\times 10^{7}\)
Root analytic conductor: \(4.55731\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{12} \cdot 17^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(10.63710802\)
\(L(\frac12)\) \(\approx\) \(10.63710802\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 \)
good2 \( 1 + 3 T + 3 p T^{2} + 11 T^{3} + 9 p T^{4} + 15 p T^{5} + 3 p^{4} T^{6} + 15 p^{2} T^{7} + 9 p^{3} T^{8} + 11 p^{3} T^{9} + 3 p^{5} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
5 \( 1 - 3 T + 12 T^{2} - 24 T^{3} + 3 p^{2} T^{4} - 213 T^{5} + 536 T^{6} - 213 p T^{7} + 3 p^{4} T^{8} - 24 p^{3} T^{9} + 12 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 - 3 T + 15 T^{2} - 30 T^{3} + 150 T^{4} - 192 T^{5} + 839 T^{6} - 192 p T^{7} + 150 p^{2} T^{8} - 30 p^{3} T^{9} + 15 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 - 9 T + 60 T^{2} - 296 T^{3} + 1269 T^{4} - 4755 T^{5} + 16356 T^{6} - 4755 p T^{7} + 1269 p^{2} T^{8} - 296 p^{3} T^{9} + 60 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 - 9 T + 87 T^{2} - 482 T^{3} + 2688 T^{4} - 10872 T^{5} + 44767 T^{6} - 10872 p T^{7} + 2688 p^{2} T^{8} - 482 p^{3} T^{9} + 87 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 - 9 T + 105 T^{2} - 648 T^{3} + 240 p T^{4} - 21474 T^{5} + 111521 T^{6} - 21474 p T^{7} + 240 p^{3} T^{8} - 648 p^{3} T^{9} + 105 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 9 T + 102 T^{2} - 558 T^{3} + 4053 T^{4} - 17037 T^{5} + 104432 T^{6} - 17037 p T^{7} + 4053 p^{2} T^{8} - 558 p^{3} T^{9} + 102 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 - 6 T + 171 T^{2} - 813 T^{3} + 12231 T^{4} - 45633 T^{5} + 470690 T^{6} - 45633 p T^{7} + 12231 p^{2} T^{8} - 813 p^{3} T^{9} + 171 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 - 24 T + 12 p T^{2} - 4226 T^{3} + 37860 T^{4} - 277500 T^{5} + 1691791 T^{6} - 277500 p T^{7} + 37860 p^{2} T^{8} - 4226 p^{3} T^{9} + 12 p^{5} T^{10} - 24 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 + 3 T + 147 T^{2} + 274 T^{3} + 9672 T^{4} + 10032 T^{5} + 414439 T^{6} + 10032 p T^{7} + 9672 p^{2} T^{8} + 274 p^{3} T^{9} + 147 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 - 18 T + 309 T^{2} - 3267 T^{3} + 32637 T^{4} - 245871 T^{5} + 1773626 T^{6} - 245871 p T^{7} + 32637 p^{2} T^{8} - 3267 p^{3} T^{9} + 309 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 141 T^{2} + 297 T^{3} + 9951 T^{4} + 33021 T^{5} + 481646 T^{6} + 33021 p T^{7} + 9951 p^{2} T^{8} + 297 p^{3} T^{9} + 141 p^{4} T^{10} + p^{6} T^{12} \)
47 \( 1 + 24 T + 453 T^{2} + 5671 T^{3} + 61935 T^{4} + 530319 T^{5} + 4048494 T^{6} + 530319 p T^{7} + 61935 p^{2} T^{8} + 5671 p^{3} T^{9} + 453 p^{4} T^{10} + 24 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 24 T + 489 T^{2} + 6455 T^{3} + 75489 T^{4} + 681369 T^{5} + 5546778 T^{6} + 681369 p T^{7} + 75489 p^{2} T^{8} + 6455 p^{3} T^{9} + 489 p^{4} T^{10} + 24 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 9 T + 294 T^{2} - 2214 T^{3} + 38787 T^{4} - 239589 T^{5} + 2942228 T^{6} - 239589 p T^{7} + 38787 p^{2} T^{8} - 2214 p^{3} T^{9} + 294 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 21 T + 411 T^{2} - 5420 T^{3} + 63636 T^{4} - 611928 T^{5} + 5190733 T^{6} - 611928 p T^{7} + 63636 p^{2} T^{8} - 5420 p^{3} T^{9} + 411 p^{4} T^{10} - 21 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 6 T + 249 T^{2} + 1819 T^{3} + 30213 T^{4} + 235809 T^{5} + 2399994 T^{6} + 235809 p T^{7} + 30213 p^{2} T^{8} + 1819 p^{3} T^{9} + 249 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 27 T + 633 T^{2} - 9886 T^{3} + 131955 T^{4} - 1415775 T^{5} + 13062822 T^{6} - 1415775 p T^{7} + 131955 p^{2} T^{8} - 9886 p^{3} T^{9} + 633 p^{4} T^{10} - 27 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 - 18 T + 408 T^{2} - 4781 T^{3} + 66651 T^{4} - 601068 T^{5} + 6253045 T^{6} - 601068 p T^{7} + 66651 p^{2} T^{8} - 4781 p^{3} T^{9} + 408 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 - 24 T + 483 T^{2} - 7512 T^{3} + 96486 T^{4} - 1056696 T^{5} + 10233479 T^{6} - 1056696 p T^{7} + 96486 p^{2} T^{8} - 7512 p^{3} T^{9} + 483 p^{4} T^{10} - 24 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 6 T + 255 T^{2} + 911 T^{3} + 33621 T^{4} + 66129 T^{5} + 3005934 T^{6} + 66129 p T^{7} + 33621 p^{2} T^{8} + 911 p^{3} T^{9} + 255 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 207 T^{2} - 585 T^{3} + 18153 T^{4} - 157635 T^{5} + 1321982 T^{6} - 157635 p T^{7} + 18153 p^{2} T^{8} - 585 p^{3} T^{9} + 207 p^{4} T^{10} + p^{6} T^{12} \)
97 \( 1 + 33 T + 906 T^{2} + 16414 T^{3} + 259359 T^{4} + 3196683 T^{5} + 35074495 T^{6} + 3196683 p T^{7} + 259359 p^{2} T^{8} + 16414 p^{3} T^{9} + 906 p^{4} T^{10} + 33 p^{5} T^{11} + p^{6} T^{12} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−4.56378336089798763783267612424, −4.45024802579595638529291345787, −4.29641439310949236547299392260, −4.24211224873688422271415065012, −3.96789378043022976715306135869, −3.74737741677403356095663851450, −3.73802778952328153197774722326, −3.44487857661070500240694711780, −3.32631415256266792416812456942, −3.16742410130211824796808909471, −3.13364398619885055508889554251, −3.02212163525981283269866790422, −2.52148021056827601507109655046, −2.50204505157291194529706694334, −2.43410695426082169271956506341, −2.04031842568915001953002625849, −1.76869874628315080142835144511, −1.65260959176254934205208701466, −1.57172453538425916947303360436, −1.34797058362322377388037443169, −1.16686439571468830900671298310, −0.954509145927745915830424954681, −0.798354625028308933122316813468, −0.66006323600139835036125663189, −0.53791134598819237984471073686, 0.53791134598819237984471073686, 0.66006323600139835036125663189, 0.798354625028308933122316813468, 0.954509145927745915830424954681, 1.16686439571468830900671298310, 1.34797058362322377388037443169, 1.57172453538425916947303360436, 1.65260959176254934205208701466, 1.76869874628315080142835144511, 2.04031842568915001953002625849, 2.43410695426082169271956506341, 2.50204505157291194529706694334, 2.52148021056827601507109655046, 3.02212163525981283269866790422, 3.13364398619885055508889554251, 3.16742410130211824796808909471, 3.32631415256266792416812456942, 3.44487857661070500240694711780, 3.73802778952328153197774722326, 3.74737741677403356095663851450, 3.96789378043022976715306135869, 4.24211224873688422271415065012, 4.29641439310949236547299392260, 4.45024802579595638529291345787, 4.56378336089798763783267612424

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.